There has been much discussion on the SETI email list of Zone Plate (reflecting) Fresnel, or "bullseye,"
antennas. A "bullseye" is American slang for a target used
in archery and shooting practice which consists
of concentric red rings surrounding a central red
disc. The Zone Plate antenna is just such a pattern,
but the red is replaced by metallic reflectors, and
the area between the red is replaced by nonreflectors.
The entire antenna is flat, that is, all reflecting and
nonreflecting "zones" are intended to be confined
to the x,y plane.

The current design calls for a central disc reflector
surrounded by seven concentric reflecting zones. The following BASIC program allows
one to predict the locations of the borders of
the zones:

Here F0 is the frequency (1420 MHz ), LA is the
wavelength, R0 is the height above ground to the
45 degree feedhorn (40 feet or 12.92 meters). And
S is the distance from the center of the bullseye to
the border, in meters.

When RUN, this program produces about 56 borders,
or about 23 coincentric bands, with alternating bands
reflecting or absorbing. Wayne S. (or Aleph Null on
SETI net) suggested placing such a reflecting array
1/4 wavelength (about two inches) above a reflecting
groundplane. This will probably work, but I have yet
to fully simulate it.

The radius of the central disc is 1.695 meters. The
seven rings are best described by their "starting" and
"ending" radii, and are (in meters):

Ring Number

Start Radius

Ending Radius

1

2.402

2.947

2

3.409

3.820

3

4.192

4.537

4

4.859

5.163

5

5.453

5.729

6

5.995

6.252

7

6.500

6.740

Up the z axis 13.55 meters is a feedhorn looking
down at the bullseye. The feedhorn has a 45 degree
reception angle, to receive energy from all the zones,
including the seventh zone. The feedhorn is located
directly above the center of the central disc reflector.

Actually, this is only the most rudimentary
beginning of the simulation which sums the
contributions from all these zones at the
feedhorn.
However, the remainder of the program will
just add up the contributions from alternate
zones, using Cos(Kr), Sin(kr) terms (K is
just wave-number K=2*pi/ LA) and the area
of each contribution (da=2*pi*s*ds). Once
the Cos terms are all added up, and the Sin
terms are added up, the net amplitude is
a=sqr(Cos^2+Sin^2), which is the "zero
order" straight-up amplitude.

The off-axis terms just add a linear phase-
term in X (or Y), if the bullseye is in the X,Y
plane.

The groundplane is NOT included in this simulation.
All dimensions are in meters, angles in radians, except
for DG which is the off-axis pointer, and is in degrees.

R0 is feedhorn height above the bullseye, and is also
approximate bullseye outer diameter. LA is wavelength.
A is the computed amplitude of the received signal. It is
expressed as linear: to "normalize" the beampattern
response, each off-axis A should be divided by the A for
zero degrees, and 20*log of this term be taken.

Lines 10-100 set up the "borders" between the Zones,
and load these radii into the array S[N], and prints out
these borders with their corresponding index N.

Lines 1000-1090 (the "Do a Band" subroutine)
does most of the work. The most nested loop rotates
the angle AD ( for Angle in Disc ) in steps of size
determined by line 1030, which is set at a step size of
1/100 wavelength. Rotation of AD through Pi radians
is sufficient, as continued rotation only duplicates.
The outer loop in "Do a Band" repeats the process for
all eight active zones of this design, skipping those
zones from which energy reaching the feedhorn is
out-of-phase. Line 1010 sets the step size in radius at
1/100 wavelength.

This program was written and debugged on an Apple II
computer in Apple Basic. The variables used in debugging
were for a 10 foot diameter antenna with only ONE zone
surrounding the central disc, and step sizes were 1/10
wavelength . This was done so that this program could be
debugged and presented to the Group within a reasonable
time. The Apple II is REALLY SLOW. We suspect ET will
come before the Apple II finished the simulation above.

O.K., you Computer Masters: rewrite this thing in "C" and
use the math coprocessors for the Sin, Cos. The simulation
is accurate, and produces the expected tight beams and
sidelobes. There may be a typo left, but I don't think so.

While I have been working on the Zone Plate
simulation, I have been aware of some of the
discussions about gratings/etc.

I AM CONCERNED that the simulated disc
may not perform as well as the simulation
predicts. If it were constructed with a "specular"
metallic surface, composed of hills/valleys on
the order of 1/4 wave high, I would expect it to
act nearly as good as the simulations predict.
But with entirely "flat" construction techniques,
I am not entirely sure. Currently we are considering perfect, flat metallic
reflectors. BUT WE WOULD be most interested in
simulation results with imperfect, or more "specular"
metallic surfaces, like an "imperfect" surface finish.

I had originally thought that, because the thinner
zones were further removed from the feedhorn,
they would "scatter" the incident RF enough so
that a large portion would enter the feedhorn. The
outer zone is only 1.1 wavelength wide, and such
a narrow metallic surface might scatter the energy
sufficiently.

Several approaches suggest themselves if it is
determined that the "specularity" of these surfaces
needs to be increased. I have seen the suggestions
of a sinusoidal surface, and noted the suggestions
of "tilted" surfaces. In optics, this is called "blazing".
An approach might be to divide each of these zones
into many smaller ( thinner, say 1 cm wide ) "minizones".
There was work done in the 1950's using foam plastics
as lenses: it was found that the increased dielectric
constant of certain foams could be used to produce
what today we would call a "phase grating".

Whatever solution works best will ultimately be determined by experimentation. And not necessarily on
large systems. Many good graduate-level EE labs
have HP microwave benches capable of measuring
the scattering angles of 21 cm radiation from test
surfaces. But as with all scientific progress, experimentation will be the key.